Moment Of Inertia

The moment of inertia about an axis passing through centre of mass of disc and
perpendicular to its plane is

where is the mass of disc and its radius. According to theorem of parallel axis, moment of inertia of circular disc about an axis touching the disc at its diameter and normal to the disc is

Answer (d)

Using the principle of conservation of angular momentum: The initial moment of inertia of the system is: The final moment of inertia after placing the second disc is: Substituting these into the conservation of angular momentum equation: Simplifying: Final Answer:

Step 1: Moment of Inertia for a Solid Sphere.
The moment of inertia of a solid sphere of mass and radius about its diameter is: Step 2: Moment of Inertia about the Given Axis (Parallel Axis Theorem).
Using the Parallel Axis Theorem, the moment of inertia about an axis parallel to the diameter and at a distance from the center is: Here: - (distance from the center of each sphere to the axis). For each sphere, the moment of inertia about the given axis is: Substitute and : Step 3: Total Moment of Inertia for Two Spheres.
Since there are two spheres and the axis passes symmetrically through the midpoint, the total moment of inertia is: Step 4: Final Answer.
The total moment of inertia of the system is:

The moment of inertia ( ) of a rectangular sheet about an axis passing through its center of mass ( ) is: where and are the dimensions of the rectangle, and is the mass of the sheet. Step 1: Calculate Substitute the given values: Simplify: Step 2: Use the Parallel Axis Theorem to Find The moment of inertia about is given by: where is the perpendicular distance between and . Calculate : Substitute : Simplify: Step 3: Find the Ratio of to Final Answer:

The rotational kinetic energy is given by: Where is the moment of inertia and is the angular velocity. ### Moment of inertia for the solid cylinder: The moment of inertia for a solid cylinder is: ### Moment of inertia for the hollow cylinder: The moment of inertia for a hollow cylinder is: Thus, the rotational kinetic energy for each cylinder is: Now, the ratio of their rotational kinetic energies is: Thus, the ratio of their rotational kinetic energies is 2.

Concept: For a ring:

• Moment of inertia about a diameter in its plane is
• To find the moment of inertia about a tangent in the same plane, we use the Parallel Axis Theorem: where is the distance between the two parallel axes.

Step 1: {Moment of inertia about diameter.}
Step 2: {Apply the Parallel Axis Theorem.} Distance between diameter and tangent Since the axis lies in the plane and passes through the tangent point, the total moment of inertia about the tangent becomes